Extend polynomial identities to complex numbers for higher-degree polynomial work.
Factor a cube polynomial over R and C and list roots
Problem
Factor \(x^{3}~-~8\) completely into linear factors over the complex numbers.
Big Picture
What this problem is really about
Use the sum- or difference-of-cubes identity to expose one real linear factor and a remaining quadratic. The real factor gives one root immediately; solve the quadratic without stopping just because it is irreducible over the reals. A negative discriminant produces a conjugate complex pair, completing the number of roots required by the cubic's degree.
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